Invariants
| Level: | $40$ | $\SL_2$-level: | $4$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.344 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&22\\32&7\end{bmatrix}$, $\begin{bmatrix}11&22\\38&19\end{bmatrix}$, $\begin{bmatrix}31&8\\4&39\end{bmatrix}$, $\begin{bmatrix}31&20\\10&13\end{bmatrix}$, $\begin{bmatrix}31&36\\14&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.c.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $15360$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 49 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{(x+y)^{24}(x^{8}+56x^{4}y^{4}+16y^{8})^{3}}{y^{4}x^{4}(x+y)^{24}(x^{2}-2y^{2})^{4}(x^{2}+2y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 20.24.0-4.b.1.2 | $20$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-8.a.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-8.a.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-4.b.1.11 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-8.b.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-8.b.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.96.0-8.d.1.5 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.d.1.5 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.e.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.e.2.5 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.e.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.e.2.5 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.f.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.f.1.6 | $40$ | $2$ | $2$ | $0$ |
| 40.96.1-8.j.1.8 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-8.p.1.7 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.v.1.3 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bb.1.3 | $40$ | $2$ | $2$ | $1$ |
| 40.240.8-40.c.1.6 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.c.1.10 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.c.1.6 | $40$ | $10$ | $10$ | $15$ |
| 120.96.0-24.d.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.d.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.e.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.e.2.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.e.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.e.2.11 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.f.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.f.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-24.v.1.13 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.v.1.13 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bb.1.13 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bb.1.16 | $120$ | $2$ | $2$ | $1$ |
| 120.144.4-24.c.1.24 | $120$ | $3$ | $3$ | $4$ |
| 120.192.3-24.f.1.21 | $120$ | $4$ | $4$ | $3$ |
| 280.96.0-56.d.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.d.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.e.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.e.2.7 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.e.1.15 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.e.2.22 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.f.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.f.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-56.v.1.14 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.v.1.25 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.bb.1.14 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.bb.1.30 | $280$ | $2$ | $2$ | $1$ |
| 280.384.11-56.c.1.34 | $280$ | $8$ | $8$ | $11$ |